On estimation of a class of efficacy-related parameters
1981; Taylor & Francis; Volume: 1981; Issue: 2 Linguagem: Inglês
10.1080/03461238.1981.10413733
ISSN1651-2030
Autores Tópico(s)Statistical Distribution Estimation and Applications
ResumoAbstract For specified functions φ and ψ and unknown distribution function F with density f, the efficacy-related parameter T(f) = ∫ φ(x)ψ(F(x))f 2(x)dx may be estimated by the sample analogue estimator T(fn ) based on an empirical density estimator fn . For {Xi } i.i.d. F and fn of the form fn (x) = n -1 , we approximate the estimation error T(fn ) - T(f) by the Gateaux derivative of the functional T(·) at the “point” f with increment fn -f. In conjunction with stochastic properties of the L 2-norm ‖fn -f‖, this approach leads to characterizations of the stochastic behavior of T(fn )-T(f). In particular, under mild assumptions on f, we obtain the rate of strong convergence T(fn )-T(f)=a.s. O(n-1/2(log n)1/2), which significantly improves previous results in the literature. Also, we establish asymptotic normality with associated Berry-Esséen rates.
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